My three favorite dictionaries all say that “times” is a preposition.

My short answer is that it is a noun. My long answer follows. The sentence has "he" as a subject, "has" as a verb, and "three times my money" as an object that completes the sentence and tells us what he has. "Three times my money" is a noun phrase. The main part "my money" is itself a noun phrase where "money" is a noun and "my" is a determiner telling us whose money. "Three times" is also a noun phrase where "three" is a number modifying the noun "times". "Three times" modifies "my money" and tells us how much money compared to just my money. So, we have "my" modifying "money"; "three" modifying "times"; "three times" modifying "my money"; and " three times my money" as an object. I had to think about this one. Language develops wildly. People want to communicate something with meaning and use a series of words to do that, even creating new words or using words in new ways. Eventually some professional people analyse the speaking and try to make sense of it. Sometimes that is not easy. Sometimes the analysts disagree. English develops by lots of people using it to say what they want and not by first setting out rules and then having everything follow those rules exactly.